
where \(t\)  denotes the time when the corresponding perturbation returns to the horizon. Eq. (REF ) shows that PBHs born before \(\mathcal {O}(10^{-4})\)  second since the hot big bang have mass less than 1 solar mass, which is less than the minimal mass of astrophysical black holes. Considering Hawking radiation [1]}, PBHs with an initial mass of less than \(10^{15} \mathrm {g}\)  have already evaporated by now, as a result, PBHs with an initial mass of more than \(10^{15} \mathrm {g}\)  and less than 1 solar mass still exist and produce various distinct signals in the present universe. In detecting sub-solar-mass PBHs, microlensing effect [2]}, [3]}, [4]}, [5]}, femtolensing effect [6]}, disruption of white dwarfs [7]}, disruption of neutron stars [8]}, [9]}, [10]} and so on put constraints on the fraction of sub-solar-mass PBHs in dark matter. Some interactions between black holes and stars can help detect sub-solar-mass PBHs such as superradiance instability around black holes [11]}, [12]}. However, these observational effects such as microlensing are extremely weak, only a few events give a weak constraint on the fraction of PBHs in the dark matter.
